1. Introduction
This paper is written in the context of the special issue “Hypercompositional Algebra and Applications” in “Mathematics” and it aims to shed light on two areas where the Hypercompositional Algebra has expanded and has interacted with them: Computer Science and Geometry.
Hypercompositional Algebra is a branch of Abstract Algebra which appeared in the 1930s via the introduction of the hypergroup.
It is interesting that the group and the hypergroup are two algebraic structures which satisfy exactly the same axioms, i.e., the associativity and the reproductivity, but they differ in the law of synthesis. In the first one, the law of synthesis is a composition, while in the second one it is a hypercomposition. This difference makes the hypergroup a much more general algebraic structure than the group, and for this reason the hypergroup has been gradually enriched with further axioms, which are either more powerful or less powerful, leading thus to a significant number of special hypergroups. Among them, there exist hypergroups that were proved to be very useful for the study of Formal Languages and Automata, as well as convexity in Euclidian vector spaces. Furthermore, based on these hypergroups, there derived other hypercompositional structures, which are equally as useful in the study of Geometries (spherical, projective, tropical, etc.) and Computer Science.
A
binary operation (or
composition)
on a non-void set
is a rule which assigns a unique element of
to each element of
. The notation
, where
are elements of
, indicates that
is the result of the operation
performed on the operands
and
. When no confusion arises, the operation symbol
may be omitted, and we write
If the binary operation is
associative, that is, if it satisfies the equality
the pair
is called
semigroup. An element
is an
identity of
if for all
,
. A triplet
is a
monoid if
is a semigroup and
is its identity. If no ambiguity arises, we can denote a semigroup
or a monoid
simply by
. A binary operation on
is
reproductive if it satisfies:
Definition 1. (Definition of the group) The pair is called group if is a non-void set and is an associative and reproductive binary operation on .
This definition does not appear in group theory books, but it is equivalent to the one mentioned in them. We introduce it here as we consider it to be the most appropriate in order to demonstrate the close relationship between the group and the hypergroup. Thus, the following two properties of the group derive from the above definition:
Property 1. In a group there is an element , called the identity element, such that for all in .
Property 2. For each element of a group there exists an element of , called the inverse of such that .
The proof of the above properties can be found in [
1,
2]. If no ambiguity arises, we may abbreviate
to
.
Example 1. (The free monoid) Often, a finite non-empty set is referred to as an alphabet. The elements of , i.e., the functions from to , are called strings (or words) of length When , then we have which is equal to . The empty set is the only string of length 0 over . This string is called the empty string and it is denoted by . If is a string of length over and , we write . The setbecomes a monoid if we defineThe identity element of this binary operation, which is called string concatenation, is . The strings of length 1 generate the monoid, since every element is a finite product of strings of length 1. The function from to which is defined byis a bijection. Thus we may identify a string , of length 1 over , with its only element . This means that we can regard the sets and as identical and consequently we may regard the elements of as words in the alphabet . It is obvious that for all there is exactly one natural number and exactly one sequence of elements of , such that .
is called the free monoid on the set (or alphabet) .
A generalization of the binary operation is the binary hyperoperation or hypercomposition on a non-void set , which is a rule that assigns to each element of a unique element of the power set of . Therefore, if are elements of , then . When there is no likelihood of confusion, the symbol can be omitted and we write . If are subsets of , then signifies the union . In particular if or , then In both cases, and have the same meaning as and respectively. Generally, the singleton is identified with its member .
Definition 2. (Definition of the hypergroup) The pair is called hypergroup, if is a non-void set and is an associative and reproductive binary hypercomposition on .
The following Proposition derives from the definition of the hypergroup:
Proposition 1. In a hypergroup the result of the hypercomposition of any two elements is non-void.
The proof of Proposition 1 can be found in [
3,
4]. If no ambiguity arises, we may abbreviate
to
. Two significant types of hypercompositions are the closed and the open ones. A hypercomposition is called
closed [
5] (or
containing [
6], or
extensive [
7]) if the two participating elements always belong to the result of the hypercomposition, while it is called
open if the result of the hypercomposition of any two elements different from each other does not contain the two participating elements.
The notion of the hypergroup was introduced in 1934 by F. Marty, who used it in order to study problems in non-commutative algebra, such as cosets determined by non-invariant subgroups [
8,
9,
10]. From Definitions 1 and 2 it is evident that both groups and hypergroups satisfy the same axioms and their only difference is that the law of synthesis of two elements is a composition in groups, while it is a hypercomposition in hypergroups. This difference makes the hypergroups much more general algebraic structures than the groups, to the extent that properties similar to the previous 1 and 2 generally cannot be proved for the hypergroups. Furthermore, in the hypergroups there exist different types of identities [
11,
12,
13]. In general, an element
is an identity if
for all
. An identity is called
scalar if
for all
, while it is called
strong if
for all
. Obviously, if the hypergroup has an identity, then the hypercomposition cannot be open.
Besides, in groups, both equations
and
have a unique solution, while, in the hypergroups, the analogous relations
and
do not have unique solutions. Thus F. Marty in [
8] defined the two induced hypercompositions (right and left division) that derive from the hypergroup’s hypercomposition:
If
is a group, then
and
. It is obvious that if “.” is commutative, then the right and the left division coincide. For the sake of notational simplicity,
or
is used to denote the right division, or right hyperfraction, or just the division in the commutative hypergroups and
or
is used to denote the left division, or left hyperfraction [
14,
15]. Using the induced hypercomposition we can create an axiom equivalent to the reproductive axiom, regarding which, the following Proposition is valid [
3,
4]:
Proposition 2. In a hypergroup H, the non-empty result of the induced hypercompositions is equivalent to the reproductive axiom.
W. Prenowitz enriched a commutative hypergroup of idempotent elements with one more axiom, in order to use it in the study of Geometry [
16,
17,
18,
19,
20,
21]. More precisely, in a commutative hypergroup H, all the elements of which satisfy the properties
and
, he introduced the
transposition axiom:
He named this new hypergroup
join space. For the sake of terminology unification, a commutative hypergroup which satisfies the transposition axiom is called
join hypergroup. Prenowitz was followed by J. Jantosciak [
20,
21,
22,
23], V. W. Bryant, R. J. Webster [
24], D. Freni, [
25,
26], J. Mittas, C. G. Massouros [
2,
27,
28,
29], A. Dramalidis [
30,
31], etc. In the course of his research, J. Jantosciak generalized the transposition axiom in an arbitrary hypergroup as follows:
and he named this hypergroup
transposition hypergroup [
15]. These algebraic structures attracted the interest of a big number of researchers, among whom there are, J. Jantosciak [
15,
22,
23], I. Cristea [
32,
33,
34,
35], P. Corsini, [
35,
36,
37,
38,
39,
40,
41,
42], V. Leoreanu-Fortea [
40,
41,
42,
43,
44,
45,
46], S. Hoskova-Mayerova, [
47,
48,
49,
50,
51], J. Chvalina [
48,
49,
50,
51,
52], P. Rackova [
49,
50], C. G. Massouros [
3,
6,
12,
13,
53,
54,
55,
56,
57,
58,
59,
60,
61,
62], G. G. Massouros [
3,
12,
13,
53,
54,
55,
56,
57,
58,
59,
60,
61,
62], R. Ameri [
63,
64,
65], M. M. Zahedi [
63], I. Rosenberg [
66], etc.
Furthermore, it has been proved that these hypergroups also comprise a useful tool in the study of Languages and Automata [
67,
68,
69,
70,
71] and a constructive origin for the development of other, new, hypercompositional structures [
53,
57,
58,
72,
73].
Definition 3. A transposition hypergroup, which has a scalar identity e, is called quasicanonical hypergroup [
74,
75]
or polygroup [
76,
77,
78].
In the quasicanonical hypergroups, there exist properties analogous to 1 and 2 which are valid in the groups:
Proposition 3. [
15,
53]
If is a quasicanonical hypergroup, then:- (i)
for each there exists one and only one such that
- (ii)
The inverse is also true:
Proposition 4. [
15,
53]
If a hypergroup Q has a scalar identity e and- (i)
for each there exists one and only one such that
- (ii)
then, the transposition axiom is valid in Q.
A commutative quasicanonical hypergroup is called
canonical hypergroup. This hypergroup was first used by M. Krasner [
79] but it owes its name to J. Mittas [
80,
81].
A non-empty subset
of a hypergroup
is called
semi-subhypergroup when it is stable under the hypercomposition, i.e.,
for all
.
is a
subhypergroup of
if it satisfies the reproductive axiom, i.e., if the equality
is valid for all
. Since the structure of the hypergroup is much more complicated than that of the group, there are various kinds of subhypergroups. A subhypergroup
of H is called
closed from the right (in H), (resp.
from the left) if, for every element
in the complement
of
, it holds that
(resp.
).
is called
closed if it is closed both, from the right and from the left [
82,
83,
84]. It has been proved [
1] that a subhypergroup is closed if and only if it is stable under the induced hypercompositions, i.e.,
A subhypergroup K of a hypergroup is invertible if implies , and implies . From this definition it derives that every invertible subhypergroup is also closed, but the opposite is not valid.
Proposition 5. [
1,
14]
If a subset K of a hypergroup H is stable under the induced hypercompositions, then K is a subhypergroup of H. Proposition 6. [
1,
14]
If K is a closed subhypergroup of a hypergroup H and , then: It has been proved [
14,
60,
62] that the set of the semi-subhypergroups (resp. the set of the closed subhypergroups) which contains a non-void subset
is a complete lattice. Hence, the minimum (in the sense of inclusion) semi-subhypergroup of a hypergroup
, which contains a given non-empty subset
of
H, can be assigned to
E. This semi-subhypergroup is denoted by
and it is called the generated by
semi-subhypergroup of
. Similarly,
is the generated by
closed subhypergroup of
. For notational simplicity, if
, then
and
are used instead.
Duality. Two statements of the theory of hypergroups are
dual statements (see [
15,
53]), if each one of them results from the other by interchanging the order of the hypercomposition “·”, that is, interchanging any hyperproduct
ab with
ba. Observe that the reproductive and the associative axioms are self-dual. Moreover, observe that the induced hypercompositions / and \ have dual definitions; hence, they must be interchanged during the construction of a dual statement. So, the transposition axiom is self-dual as well. Therefore, the following principle of duality holds for the theory of hypergroups:
Given a Theorem, the dual statement, which results from the interchange of the order of the hypercomposition (and the necessary interchange of / and \), is also a Theorem.
Special notation: In the following pages, apart from the typical algebraic notations, we are using Krasner’s notation for the complement and difference. So, we denote with A..B the set of elements that are in the set A, but not in the set B.
2. Formal Languages, Automata Theory and Hypercompositional Structures
Mathematically, a language whose words are written with letters from an alphabet Σ, is defined as a subset of the free monoid Σ* generated by Σ. The above definition of the language is fairly general and it includes all the written natural languages as well as the artificial ones. In general, a language is defined in two ways: It is either presented as an exhaustive list of all its valid words, i.e., through a dictionary, or it is presented as a set of rules defining the acceptable words. Obviously the first method can only be used when the language is finite. All the natural languages, such as English or Greek are finite and they have their own dictionaries. Artificial languages, on the other hand, may be infinite, and they can only be defined by the second way.
In the artificial languages, precision and no guesswork are required, especially when computers are concerned. The regular expressions, which are very precise language-defining symbols, were created and developed as a language-defining symbolism. The languages that are associated with these expressions are called regular languages.
The regular expressions were introduced by Kleene [
85] who also proved that they are equivalent in expressive power to finite automata. McNaughton and Yamada gave their own proof to this [
86], while Brzozowski [
87,
88] and Brzozowski and McClusky [
89] further developed the theory of regular expressions. In regular languages the expression
where
and
are strings of characters from an alphabet
means “either
or
”. Therefore
. In this way the monoid
is enriched with a hypercomposition. This hypercomposition is named
B-hypercomposition [
67,
68,
69].
Proposition 7. [
67,
68]
A non-void set equipped with the B-hypecomposition is a join hypergroup. A hypergroup equipped with the B-hypercomposition is called
B-hypergroup [
67,
68,
69]. Moreover, the empty set of words and its properties in the theory of the regular languages leads to the following extension: Let
. In the set
a hypercomposition, called
dilated B-hypercomposition, is defined as follows:
The associativity and the commutativity of the dilated B-hypercomposition derive without difficulty. Moreover, the transposition axiom is verified, since
This join hypergoup is called dilated B-hypergroup and it has led to the definition of a new class of hypergroups, the class of the fortified transposition hypergroups and fortified join hypergroups.
An automaton is a collection of five objects , where is the alphabet of input letters (a finite nonempty set of symbols), is a finite nonvoid set of states, is an element of indicating the start (or initial) state, is a (possibly empty) subset of representing the set of the final (or accepting) states and is the state transition function with domain and range , in the case of a deterministic automaton (DFA), or , the powerset of , in the case of a nondeterministic automaton (NDFA). denotes the set of words (or strings) formed by the letters of –closure of – and signifies the empty word. Given a DFA , the extended state transition function for , denoted , is a function with domain and range defined recursively as follows:
- i.
for all in and in
- ii.
for all in
- iii.
for all in , in and in .
In [
67,
68,
69,
70,
71] it is shown that the set of the states of an automaton, equipped with different hypercompositions, can be endowed with the structure of the hypergroup. The hypergroups that have derived in this way are named
attached hypergroups to the automaton. To date, various types of attached hypergroups have been developed to represent the structure and operation of the automata with the use of the hypercompositional algebra tools. Between them are:
- i.
the attached hypergroups of the order, and
- ii.
the attached hypergroups of the grade.
Those two types of hypergroups were also used for the minimisation of the automata. In addition, in [
69] another hypergroup, derived from a different definition of the hypercomposition, was attached to the set of the states of the automaton. Due to its definition, this hypergroup was named the
attached hypergroup of the paths and it led to a new proof of Kleene’ s Theorem. Furthermore, in [
70], the
attached hypergroup of the operation was defined in automata. One of its applications is that this hypergroup can indicate all the states on which an automaton can be found after the t-clock pulse. For the purpose of defining the attached hypergroup of the operation, the notions of the Prefix and the Suffix of a word needed to be introduced. Let
x be a word in Σ
*, then:
Let
s be an element of
S. Then:
Obviously, the states
and
are in
.
Lemma 1. If , then .
Proof. and since , it holds that , for some in . Thus and so the Lemma. □
With the use of the above notions, more hypercompositional structures can be attached on the set of the states of the automaton.
Proposition 8. The set S of the states of an automaton equipped with the hypercompositionbecomes a join hypergroup. Proof. Initially, notice that
. Hence, the reproductive axiom is valid. Next, the definition of the hypercomposition yields the equality:
Per Lemma 1, the right-hand side of the above equality is equal to
which, however, is equal to
Using again Lemma 1, we get the equality
Thus:
and so, the associativity is valid. Next, observe that the hypercomposition is commutative and therefore:
Suppose that . Then, which is non-empty, since it contains so. Hence the transposition axiom is valid and so the Proposition. □
Proposition 9. The set S of the states of an automaton equipped with the hypercompositionbecomes a join semihypergroup. Proof. Since
, for all
, the result of the hypercomposition is always non-void. On the other hand
hence, since
, with
s,
q in
S, is not always nonvoid, the reproductive axiom is not valid. The associativity can be verified in the same way as in the previous Proposition. Finally if
, then
and so the intersection
which is equal to
is non-empty, since it contains
. □
G. G. Massouros [
68,
69,
70,
71,
72,
73], G. G. Massouros and J. D. Mittas [
67] and after them J. Chvalina [
90,
91,
92], L. Chvalinova [
90], M. Novak [
91,
92,
93,
94], S. Křehlík [
91,
92,
93], M. M. Zahedi [
95], M. Ghorani [
95,
96] etc, studied automata using algebraic hypercompositional structures.
Formal Languages and Automata theory are very close to Graph theory. P. Corsini [
97,
98], M. Gionfriddo [
99], Nieminen [
100,
101], I. Rosenberg [
66], M. De Salvo and G. Lo Faro [
102,
103,
104], I. Cristea et al. [
105,
106,
107,
108], C. Massouros and G. Massouros [
109,
110], C. Massouros and C. Tsitouras [
111,
112] and others studied hypergroups associated with graphs. In the following we will present how to attach a join hypergroup to a graph. In general, a
graph is a set of points called
vertices connected by lines, which are called
edges. A
path in a graph is a sequence of no repeated vertices
, such that
are edges in the graph. A graph is said to be
connected if every pair of its vertices is connected by a path. A
tree is a connected graph with no cycles. Let
be a tree. In the set
of its vertices a hypercompostion “⋅“ can be introduced as follows: for each two vertices
in
,
is the set of all vertices which belong to the path that connects vertex
with vertex
. Obviously this hypercomposition is a closed hypercomposition, i.e.,
are contained in
for every
in
.
Proposition 10. If V is the set of the vertices of a tree , then (V, ⋅) is a join hypergroup.
Proof. Since , it derives that for each in and therefore the reproductive axiom is valid. Moreover, since is an undirected graph, the hypecomposition is commutative. Next, let be three arbitrary vertices of . If any of these three vertices, e.g., z, belongs to the path that the other two define, then . If do not belong to the same path, then there exists only one vertex in such that . Indeed if there existed a second vertex such that , then the tree would have a cycle, which is absurd. So and . Since , it derives that . Now, for the transposition axiom, suppose that are vertices of such that . If are in the same path, then considering all their possible arrangements in their path, it derives that . Next, suppose that the four vertices do not belong to the same path. Thus, suppose that does not belong to the path defined by . Then, . Consider and . As indicated above, since there are no cycles in , there exists only one vertex in such that and . Now, we distinguish between the cases:
- (i)
if do not belong to the same path, then for the same reasons as above there exists only one in such that and . Since , there exists in such that and . Thus, since contains no cycles, and in order for not to form a cycle, and must coincide. Hence, and therefore .
- (ii)
if belongs to the same path with and , then:
- (iia)
if , then and . Hence, , and therefore .
- (iib)
if , then . □
A spanning tree of a connected graph is a tree whose vertex set is the same as the vertex set of the graph, and whose edge set is a subset of the edge set of the graph. Any connected graph has at least one spanning tree and there exist algorithms, which find such trees. Hence, any connected graph can be endowed with the join hypergroup structure through its spanning trees. Moreover, since a connected graph may have more than one spanning trees, more than one join hypergroups can be associated to a graph. On the other hand, in any connected or not connected graph, a hypergroup can be attached according to the following Proposition:
Proposition 11. The set of the vertices of a graph, is equipped with the structure of the hypergroup, if the result of the hypercomposition of two vertices and is the set of the vertices which appear in all the possible paths that connect to , or the set , if there do not exist any connecting paths from vertex to vertex .
2.1. Fortified Transposition Hypergroups
Definition 4. A fortified transposition hypergroup (FTH) is a transposition hypergroup H with a unique strong identity e, which satisfies the axiom:
for every there exists one and only one element , such that and .
is denoted by and it is called inverse or symmetric of . When the hypercomposition is written additively, the strong identity is denoted by 0, the unique element is called opposite or negative instead of inverse and the notation is used. If the hypercomposition is commutative, the hypergroup is called fortified join hypergroup (FJH).
It has been proved that every FTH consists of two types of elements, the
canonical (
c-elements) and the
attractive (
a-elements) [
53,
57]. An element
is called canonical if
is the singleton
, while it is called attractive if
. We denote with
the set of the a-elements and with
the set of the c-elements. By convention
.
Proposition 12. - (i)
if x is a non-identity attractive element, then
- (ii)
if x is a canonical element, then
- (iii)
if x, y are attractive elements and , then and
- (iv)
if x, y are canonical elements, then and .
Theorem 1. - (i)
the result of the hypercomposition of two a-elements is a subset of A and it always contains these two elements.
- (ii)
the result of the hypercomposition of two non-symmetric c-elements consists of c-elements,
- (iii)
the result of the hypercomposition of two symmetric c-elements contains all the a-elements.
- (iv)
the result of the hypercomposition of an a-element with a c-element is the c-element.
Theorem 2. [
53,
57]
If H is a FTH, then the set A of the attractive elements is the minimum (in the sense of inclusion) closed subhypergroup of H. The proof of the above Theorems as well as other properties of the theory of the FΤHs and FJHs can be found in [
53,
55,
56,
57,
61]. The next two Propositions refer to the reversibility in FTHs.
Lemma 2. If , then and
Proof. implies and Moreover and Consequently, and . Next, the transposition axiom gives the Lemma. □
Proposition 13. If , and if any one of is a canonical element, then Proof. We distinguish between two cases:
- (i)
If , then and . Next Lemma 2 applies and yields the Proposition.
- (ii)
Suppose that and . Then, according to Theorem 1(iv), ; thus, . Via Theorem 1(iii), ; thus, . Per Theorem 2, , consequently .
Hence the Proposition is proved. □
Proposition 14. Suppose that are attractive elements and .
- (i)
if , then implies that , while
- (ii)
if , then implies that , while, generally,
- (iii)
in any other case implies and
Sketch of Proof. Cases (i) and (ii) are direct consequences of the Theorem 1, while case (iii) derives from the application of Lemma 2. □
The property which is described in Proposition 13 is called reversibility and because of Proposition 14, this property holds partially in the case of TPH.
Another distinction between the elements of the FTH stems from the fact that the equality
(or
in the additive case) is not always valid. The elements that satisfy the above equality are called
normal, while the others are called
abnormal [
53,
57].
Example 2. Let H be a totally ordered set, dense and symmetric around a center denoted by With regards to this center the partition can be defined, according to which, for every and it is and for every , where –x is the symmetric of x with regards to 0. Then H, equipped with the hypercomposition:andbecomes a FJH in which , for every .
Proposition 15. The canonical elements of a FTH are normal.
Proof. Let
be a canonical element. Because of Theorem 1,
while, according to Theorem 2,
. Thus
and therefore
. Suppose that
z is a canonical element in
. Per Proposition 13,
. So
. Hence, we have the sequence of implications:
So,
. Furthermore
implies that
and therefore the Proposition holds. □
An important Theorem that is valid for TFH [
53] is the following structure Theorem:
Theorem 3. A transposition hypergroup H containing a strong identity e is isomorphic to the expansion of the quasicanonical hypergroup by the transposition hypergroup A of all attractive elements through the identity e.
The special properties of the FTH give different types of subhypergroups. There exist subhypergroups of a FTH that do not contain the symmetric of each one of their elements, while there exist others that do. This leads to the definition of the symmetric subhypergroups. A subhypergroup
of a FTH is
symmetric, if
implies
. It is known that the intersection of two subhypergroups is not always a subhypergroup. In the case of the symmetric subhypergroups though, the intersection of two such subhypergroups is always a symmetric subhypergoup [
57,
62]. Therefore, the set of the symmetric subhypergroups of a FTH consist a complete lattice. It is proved that the lattice of the closed subhypergroups of a FTH is a sublattice of the lattice of the symmetric subhypergroups of the FTH [
57,
62]. An analytic and detailed study of these subhypergroups is provided in the papers [
1,
60,
62]. Here, we will present the study of the
monogene symmetric subhypergroups, i.e., symmetric subhypergroups generated by a single element. So, let
be a FJH, let
be an arbitrary element of
and let
be the monogene symmetric subhypergroup which is generated by this element. Then:
and:
From the above, it derives that:
Theorem 4. If x is an arbitrary element of a FJH, then the monogene symmetric subhypergroup which is generated by this element is: Proof. The symmetric subhypergroup of a normal FTH which is generated from a non-empty set
consists of the unions of all the finite products of the elements that are contained in the union
[
62]; thus, from (1) we have:
According to (2), it is . But ; therefore, the Theorem is established. □
From the above Theorem, Proposition 13 and Theorem 1, we have the following Corollary:
Corollary 1. Every monogene symmetric subhypergroup M(x) with generator a canonical element x is closed, it contains all the attractive elements and also Remark 1. - (i)
Since the inclusion is valid for .
- (ii)
for , it is .
Let us define now a symbol (which can even be the +∞), and name it order of and simultaneously order of the monogene subhypergroup . Two cases can appear such that one revokes the other:
I. For any
, with
, we have:
Then we define the order of
and of
to be the infinity and we write
.
Proposition 16. If , then x is a canonical element.
Proof. Suppose that
belongs to the set
of the attractive elements. Then, per Theorem 1.(i),
and
. Consequently:
This contradicts our assumption and therefore
is a canonical element. □
The previous Proposition and Theorem 1 result to the following Corollary:
Corollary 2. If , then does not contain attractive elements for every .
Proposition 17. If , thenfor any , with .
Proof. From it derives that . According to Proposition 16, is a canonical element and therefore, because of Proposition 15, is normal; thus, . Therefore or . So, the Proposition follows from the reversibility. □
Proposition 18. , if and only if
- (i)
, for every
- (ii)
, for every with .
Proof. If
, per Corollary 2,
does not contain attractive elements for every
Moreover, if
, then (ii) derives from Proposition 17. If
, then
and assuming that
with
, we successively have:
Conversely now. If for every
, the intersection
is void and if for every
with
, the intersection
is also void, then
and therefore:
and
Thus,
So, the Proposition holds. □
II. There exist
with
such that:
Proposition 19. Let be the minimum positive integer for which there exists such that . Then for a given there exist such that if and only if is divided by .
Proof. Let
,
. From
it derives that
Therefore, the Proposition.
Conversely now. If
is an a-element, then
for every
, so
, and thus the Proposition. Next, if
is a c-element, and
with
,
,
. Then:
According to our hypothesis
. Moreover, per Theorem 1, the sum of two non-opposite c-elements does not contain any a-elements. Consequently, there do not exist a-elements in
, and so
Thus
and therefore
This contradicts the supposition, according to which is the minimum element with the property . Thus , and so . □
For let be the minimum non-negative integer for which . Thus a function is defined which corresponds each in to the non-negative integer .
Definition 5. The pair is called order of and of . The number is called principal order of and of , while the function q is called associated order of and of .
Consequently, according to the above definition, if is an attractive element, then and therefore with for every . Moreover, if is a self-inverse canonical element, then , if and , if and thus with in the first case and with in the second case (for every ).
Moreover, we remark that the order of is , with , for every , and is the only element which has this property. Yet, it is possible that there exist non-identity elements with prime order 1, and this happens if and only if there exists an integer such that , as for example when is a self-inverse canonical element.
2.2. The Hyperringoid
Let Σ* be the set of strings over an alphabet Σ. Then:
Proposition 20. String concatenation is distributive over the B-hypecomposition.
Proof. Let . Then, . □
Via the thorough verification of the distributive axiom in all the different cases and taking into consideration that 0 is a bilaterally absorbing element with respect to the string concatenation on the set , it can also be proved that:
Proposition 21. String concatenation is distributive over the dilated B-hypecomposition.
Consequently, and are algebraic structures equipped with a composition and a hypercomposition which are related with the distributive law.
Definition 6. A hyperringoid is a non-empty set Y equipped with an operation “⋅” and a hyperoperation “+” such that:
- i.
(Y,+) is a hypergroup
- ii.
(Y, ⋅) is a semigroup
- iii.
the operation “⋅” distributes on both sides over the hyperoperation “+ ”.
If the hypergroup
has extra properties, which make it a special hypergroup, it gives birth to corresponding special hyperringoids. So, if
is a join hypergroup, then the hyperringoid is called
join. A distinct join hyperringoid is the
B-hyperringoid, in which the hypergroup is a B-hypergroup. A
fortified join hyperringoid or
join hyperring is a hyperringoid whose additive part is a fortified join hypergroup and whose zero element is bilaterally absorbing with respect to the multiplication. A special join hyperring is the
join B-hyperring, in which the hypergroup is a dilated B-hypergroup. If the additive part of a fortified join hyperringoid becomes a canonical hypergroup, then it is called
hyperring. The hyperrigoid was introduced in 1990 [
67] as the trigger for the study of languages and automata with the use of tools from hypercompositional algebra. An extensive study of the fundamental properties of hyperringoids can be found in [
61,
113,
114,
115,
116].
Example 3. Let be a ring. If in we define the hypercomposition:
then is a join hyperring. Example 4. Let be a linear order (also called a total order or chain) on Y, i.e., a binary reflexive and transitive relation such that for all either or is valid. For , the set is denoted by and the set is denoted by . The order is dense if no is void. Suppose that is a totally ordered group, i.e., is a group such that for all and , it holds that and . If the order is dense, then the set Y can be equipped with the hypercomposition:and the triplet becomes a join hyperringoid. Indeed, since the equalitiesandare valid for every , the hypercomposition is commutative and associative. Moreover,Thus, when the intersection is non-void, the intersection is also non-void. So, the transposition axion is valid. Therefore is a join hypergroup. Moreover,It is worth mentioning that the hypercomposition:endows with the join hyperringoid structure as well. As per Proposition 20, the set of the words over an alphabet can be equipped with the structure of the B-hyperringoid. This hyperringoid has the property that each one of its elements, which are the words of the language, has a unique factorization into irreducible elements, which are the letters of the alphabet. So, this hyperringoid has a finite prime subset, that is a finite set of initial and irreducible elements, such that each one of its elements has a unique factorization with factors from this set. In this sense, this hyperringoid has a property similar to the one of the Gauss’ rings. Moreover, because of Proposition 21, can be equipped with the structure of the join B-hyperring which has the same property.
Definition 7. A linguistic hyperringoid (resp. linguistic join hyperring) is a unitary B-hyperringoid (resp. join B-hyperring) which has a finite prime subset and which is non-commutative for .
It is obvious that every B-hyperringoid or join B-hyperring is not a linguistic one.
Proposition 22. From every non-commutative free monoid with finite base, there derives a linguistic hyperringoid.
Example 5. Let express the set of 2 × 2 matrices, which consist of the elements 0,1, that is the following 16 matrices: Consider the set of all 2 × 2 matrices deriving from products of the above matrices, except the zero matrix. becomes a B-hyperringoid under B-hypercomposition and matrix multiplication. Observe that none of the matrices can be written as the product of any two matrices from the set while all the matrices in result from products of these three matrices. Therefore, T is a linguistic hyperringoid, whose prime subset is .
Furthermore, if T is enriched with the zero matrix, then it becomes a linguistic join hyperring. M. Krasner was the first one who introduced and studied hypercompositional structures with an operation and a hyperoperation. The first structure of this kind was the hyperfield, an additive-multiplicative hypercompositional structure whose additive part is a canonical hypergroup and the multiplicative part a commutative group. The hyperfield was introduced by M. Krasner in [
79] as the proper algebraic tool in order to define a certain approximation of complete valued fields by sequences of such fields. Later on, Krasner introduced the hyperring which is related to the hyperfield in the same way as the ring is related to the field [
117]. Afterwards, J. Mittas introduced the
superring and the
superfield, in which both the addition and the multiplication are hypercompositions and more precisely, the additive part is a canonical hypergroup and the multiplicative part is a semi-hypergroup [
118,
119,
120]. In the recent bibliography, a structure whose additive part is a hypergroup and the multiplicative part is a semi-group is also referred to with the term additive hyperring and similarly, the term multiplicative hyperring is used when the multiplicative part is a hypergroup.
Rings and Krasner’s hyperrings have many common elementary algebraic properties, e.g., in both structures the following are true:
- (i)
- (ii)
- (iii)
In the hyperringoids though, these properties are not generally valid, as it can be seen in the following example:
Example 6. Let S be a multiplicative semigroup having a bilaterally absorbing element 0. Consider the set:With the use of the hypercomposition “+”:P becomes a fortified join hypergroup with neutral element . If is denoted by and by , then the opposite of is . Obviously this hypergroup has not c-elements. Now let us introduce in P a multiplication defined as follows:This multiplication makes a join hyperring, in whichSimilarly, .
FurthermoreMore examples of hyperringoids can be found in [
61,
113,
114,
115,
116].
3. Hypercompositional Algebra and Geometry
It is very well known that there exists a close relation between Algebra and Geometry. So, as it should be expected, this relation also appears between Hypercompositional Algebra and Geometry. It is really of exceptional interest that the axioms of the hypergroup are directly related to certain Euclid’s postulates [
121]. Indeed, according to the first postulate of Euclid:
“Hιτήσθω ἀπό παντός σημείου ἐπί πᾶν σημεῖον εὐθεῖαν γραμμήν ἀγαγεῖν” [
121]
(Let the following be postulated: to draw a straight line from any point to any point [
122])
So, to any pair of points , the segment of the straight line can be mapped. This segment always exists and it is a nonempty set of points. In fact, it is a multivalued result of the composition of two elements. Thus, a hypercomposition has been defined in the set of the points. Next, according to the second postulate:
“Καὶ πεπερασμένην εὐθεῖαν κατά τό συνεχές ἐπ’ εὐθείας ἐκβαλεῖν” [
121]
(Τo produce a finite straight line continuously in a straight line [
122])
The sets and are nonempty. Therefore, as per Proposition 2, the reproductive axiom is valid. Besides, it is easy to prove that the associativity holds in the set of the points. It is only necessary to keep in mind the definition of the equal figures given by Euclid in the “Common Notions”:
“Τὰ τῷ αὐτῷ ἲσα καὶ ἀλλήλοις ἐστίν ἲσα” [
121]
(Things which are equal to the same thing are also equal to one another [
122])
So, the set of the points is a hypergroup. Moreover, through similar reasoning, it can be proved that any Euclidean space of dimension can become a hypergroup. Indeed:
Proposition 23. Let be α linear space over an ordered field . Then , with the hypercomposition: becomes α join hypergroup. This hypergroup is called attached hypergroup. Properties of vector spaces can be found via the attached hypergroup. Thus, for example:
Proposition 24. In α vector space V over an ordered field F, the elements are affinely dependent if and only if there exist distinct integers that belong to such that: In fact, several hypergroups can be attached to a vector space [
28]. The connection of hypercompositional structures with Geometry was initiated by W. Prenowitz [
16,
17,
18,
19]. The classical geometries, descriptive geometries, spherical geometries and projective geometries can be treated as certain kinds of hypergoups, all satisfying the transposition axiom. The hypercomposition plays the central role in this approach. It assigns the appropriate connection between any two distinct points. Thus, in Euclidian geometry, it gives the points of the segment; in spherical geometry, it gives the points of the minor arc of the great circle; in projective geometry it gives the point of the line. This development is dimension free and it is applicable to spaces of arbitrary dimension, finite or infinite.
3.1. Hypergroups and Convexity
Several geometric notions can be described with the use of the hypercomposition. One such notion is the convexity. It is known that a figure is called
convex, if the segment joining any pair of its points lies entirely in it. As mentioned above, the set of the points of the plane, as well as the set of the points of any vector space
over an ordered field, becomes a hypergroup under the hypercomposition defined in Proposition 23. From this point of view, that is, with the use of the hypercomposition, a subset
of
is convex if
, for all
. However, a subset
of a hypergroup which has this property is a semi-subhypergroup [
14]. Thus:
Proposition 25. The convex subsets of a vector space V are the semi-subhypergroups of its attached hypergroup.
Consequently, the properties of the convex sets of a vector space are simple applications of the properties of the semi-subhypergroups, or the subhypergroups of a hypergroup, and more precisely, the attached hypergroup. So, this approach, except from the fact that it leads to remarkable results, it also gives the opportunity to generalize the already known Theorems of the vector spaces in sets with fewer axioms than the ones of the vector spaces. Next, we will present some well-known named Theorems that arise as corollaries of more general Theorems which are valid in hypercompositional algebra.
In hypergroups the following Theorem holds [
2,
14]:
Theorem 5. Let be α hypergroup in which every set with cardinality greater than n has two disjoint subsets such that . If with is α finite family of semi-subhypergroups of H, in which the intersection of every n elements is non-void, then all the sets have α non-void intersection.
The combination of Propositions 24, 25 and Theorem 5 gives the corollary:
Corollary 3. (Helly’s Theorem). Let be a finite family of convex sets in , with . Then, if any of the sets have a non-empty intersection, all the sets have a non-empty intersection.
Next, the following Theorem stands for a join hypergroup:
Theorem 6. Let be two disjoint semi-subhypergroups in a join hypergroup and let be an idempotent element not in the union . Then or
The proof of Theorem 6 is found in [
2,
14] and it is repeated here for the purpose of demonstrating the techniques which are used for it.
Proof. Suppose that and . Since is idempotent the equalities and are valid. Thus, there exist and , such that and . Hence, and . Thus, . Next, the application of the transposition axiom, gives . However, and , since are semi-subhypergroups. Therefore, , which contradicts the Theorem’s assumption. □
Corollary 4. Let be a join hypergroup endowed with an open hypercomposition. If are two disjoint semi-subhypergroups of and is an element not in the union , then:
The attached hypergroup of a vector space, which is defined in Proposition 23, is a join hypergroup whose hypercomposition is open, so Corollary 4 applies to it and we get the Kakutani’s Lemma:
Corollary 5. (Kakutani’s Lemma). If are disjoint convex sets in a vector space and is a point not in their union, then either the convex envelope of and or the convex envelope of and are disjoint.
Next in [
2] it is proved that the following Theorem is valid:
Theorem 7. Let be a join hypergroup consisting of idempotent elements and suppose that are two disjoint semi-subhypergroups in . Then, there exist disjoint semi-subhypergroups such that , and .
A direct consequence of Theorem 7 is Stone’s Theorem:
Corollary 6. (Stone’s Theorem). If are disjoint convex sets in a vector space , there exist disjoint convex sets and , such that , and .
During his study of Geometry with hypercompositional structures, W. Prenowitz introduced the
exchange spaces which are join spaces satisfying the axiom:
The above axiom enabled Prenowitz to develop a theory of linear independence and dimension of a type familiar to classical geometry. Moreover, a generalization of this theory has been achieved by Freni, who developed the notions of independence, dimension, etc., in a hypergroup
that satisfies only the axiom:
Freni called these hypergroups
cambiste [
25,
26]. A subset
of a hypergroup
is called
free or
independent if either
, or
for all
, otherwise it is called
non-free or
dependent.
generates
, if
, in which case
is a set of generators of
. A free set of generators is a
basis of
. Freni proved that all the bases of a cambiste hypergroup have the same cardinality. The
dimension of a cambiste hypergroup
is the cardinality of any basis of
. The dimension theory gives very interesting results in convexity hypergroups. A
convexity hypergroup is a join hypergroup which satisfies the axioms:
- i.
the hypercomposition is open,
- ii.
implies or or .
Prenowitz, defined this hypercompositional structure with equivalent axioms to the above, named it
convexity space and used it, as did Bryant and Webster [
24], for generalizing some of the theory of linear spaces. In [
2] it is proved that every convexity hypergroup is a cambiste hypergroup. Moreover in [
2] it is proved that the following Theorem stands for convexity hypergroups:
Theorem 8. Every n+1 elements of a n-dimensional convexity hypergroup are correlated.
One can easily see that the attached hypergroup of a vector space is a convexity hypergroup and, moreover, if the dimension of the attached hypergroup of a vector space is , then the dimension of is . Thus, we have the following corollary of Theorem 8:
Corollary 7. (Radon’s Theorem). Any set of d+2 points in can be partitioned into two disjoint subsets, whose convex hulls intersect.
Furthermore, the following Theorem is proved in [
2]:
Theorem 9. If is an element of a n-dimensional convexity hypergroup and are n+1 elements of such that , then there exists a proper subset of these elements which contains in their hyperproduct.
A direct consequence of this Theorem is Caratheodory’s Theorem:
Corollary 8. (Caratheodory’s Theorem). Any convex combination of points in is a convex combination of at most d+1 of them.
In addition, Theorems of the hypercompositional algebra are proved in [
2], which give as corollaries generalizations and extensions of Caratheodory’s Theorem.
An element
of a semi-subhypergroup
is called
interior element of
if for each
, there exists
, such that
. In [
2] it is proved that any interior element of a semi-subhypergroup
of a n-dimensional convexity hypergroup, is interior to a finitely generated semi-subhypergroup of
. More precisely, the following Theorem is valid [
2]:
Theorem 10. Let be an interior element of a semi-subhypergroup of a n-dimensional convexity hypergroup . Then is interior element of a semi-subhypergroup of , which is generated by at most elements.
A corollary of this Theorem, when is , is Steinitz’s Theorem:
Corollary 9. (Steinitz’s Theorem). Any point interior to the convex hull of a set in is interior to the convex hull of a subset of , containing points at the most.
D. Freni in [
123] extended the use of the hypergroup in more general geometric structures, called geometric spaces. A
geometric space is a pair
such that
is a non-empty set, whose elements are called points, and
is a non-empty family of subsets of
, whose elements are called blocks. Freni was followed by S. Mirvakili, S.M. Anvariyeh and B. Davvaz [
124,
125].
3.2. Hyperfields and Geometry
As it is mentioned in the previous
Section 2.2, the hyperfield was introduced by M. Krasner in order to define a certain approximation of a complete valued field by a sequence of such fields [
79]. The construction of this hyperfield, which was named by Krasner himself
residual hyperfield, is also described in his paper [
117].
Definition 8. A hyperring is a hypercompositional structure , where is a non-empty set, “” is an internal composition on , and “” is a hypercomposition on . This structure satisfies the axioms:
- i.
is a canonical hypergroup,
- ii.
is a multiplicative semigroup in which the zero element 0 of the canonical hypergroup is a bilaterally absorbing element,
- iii.
the multiplication is distributive over the hypercomposition (hyperaddition), i.e.,for all .
If is a multiplicative group then is called hyperfield.
J. Mittas studied these hypercompositional structures in a series of papers [
126,
127,
128,
129,
130,
131,
132,
133]. Among the plenitude of examples which are found in these papers, we will mention the one which is presented in the first paragraph of [
130].
Example 7. Let (E,⋅) be a totally ordered semigroup, having a minimum element 0, which is bilaterally absorbing with regards to the multiplication. The following hypercomposition is defined on E:Then (E,+,⋅) is a hyperring. If E..{0} is a multiplicative group, then (E,+,⋅) is a hyperfield. We referred to Mittas’ example, because nowadays, this particular hyperfield is called
tropical hyperfield (see, e.g., [
134,
135,
136,
137,
138]) and it is proved to be an appropriate and effective algebraic tool for the study of tropical geometry.
M. Krasner worked on the occurrence frequency of such structures as the hyperrings and hyperfields and he generalized his previous construction of the residual hyperfields. He observed that, if
is a ring and
is a normal subgroup of
’s multiplicative semigroup, then the multiplicative classes
, form a partition of
and that the product of two such classes, as subsets of
, is a class
as well, while their sum is a union of such classes. Next, he proved that the set
of these classes becomes a hyperring, if the product of
‘s two elements is defined to be their set-wise product and their sum to be the set of the classes contained in their set-wise sum [
117]:
and
He also proved that if is a field, then is a hyperfield. Krasner named these hypercompositional structures quotient hyperring and quotient hyperfield, respectively.
In the recent bibliography, there appear hyperfields with different and not always successful names, all of which belong to the class of the quotient hyperfields. For instance:
- (a)
starting from the papers [
139,
140] by A. Connes and C. Consani, there appeared many papers (e.g., [
135,
136,
137,
138]) which gave the name «
Krasners’ hyperfield» to the hyperfield which is constructed over the set {0, 1} using the hypercomposition:
Oleg Viro, in his paper [
135] is reasonably noticing that «
To the best of my knowledge, K did not appear in Krasner’s papers». Actually, this is a quotient hyperfield. Indeed, let
be a field and let
be its multiplicative subgroup. Then the quotient hyperfield
is isomorphic to the hyperfield considered by A. Connes and C. Consani. Hence the two-element non-trivial hyperfield is isomorphic to a quotient hyperfield.
- (b)
Papers [
139,
140] by A. Connes and C. Consani, show the construction of the hyperfield, which is now called «
sign hyperfield» in the recent bibliography, over the set {−1, 0, 1} with the following hypercomposition:
However, this hyperfield is a quotient hyperfield as well. Indeed, let
be an ordered field and let
be its positive cone. Then the quotient hyperfield
is isomorphic to the hyperfield which is called sign hyperfield.
- (c)
The so called «
phase hyperfield» (see e.g., [
135,
136]) in the recent bibliography, is just the quotient hyperfield
, where
is the field of complex numbers and
the set of the positive real numbers. The elements of this hyperfield are the rays of the complex field with origin the point (0,0). The sum of two elements
of
with
is the set
, which consists of all the interior rays
of the convex angle which is created from these two elements, while the sum of two opposite elements gives the participating elements and the zero element. This hyperfield is presented in detail in [
141].
Krasner, immediately realized that if all hyperrings could be isomorphically embedded into quotient hyperrings, then several conclusions of their theory could be deduced in a very straightforward manner, through the use of the ring theory. So, he raised the question whether all the hyperrings are isomorphic to subhyperrings of quotient hyperrings or not. He also raised a similar question regarding the hyperfields [
117]. These questions were answered by C. Massouros [
142,
143,
144] and then by A. Nakassis [
145], via the following Theorems:
Theorem 11. [
142,
143]
Let be a multiplicative group. Let , where 0 is a multiplicatively absorbing element. If H is equipped with the hypercomposition:then, is a hyperfield, which does not belong to the class of quotient hyperfields when is a periodic group. Theorem 12. [
144]
Let be the direct product of the multiplicative groups and , where . Moreover, let be the union of with the multiplicatively absorbing element 0. If K is equipped with the hypercomposition:then, is a hyperfield which does not belong to the class of quotient hyperfields when is a periodic group. Proposition 26. [
145]
Let be a multiplicative group of m, m > 3 elements. Let , where 0 is a multiplicatively absorbing element. If T is equipped with the hypercomposition:then, is a hyperfield. Theorem 13. [
145]
If is a finite multiplicative group of elements and if the hyperfield is isomorphic to a quotient hyperfield , then is a field of m-1 elements while is a field of elements. Clearly, we can choose the cardinality of
in such a way that
cannot be isomorphic to a quotient hyperfield. In [
144,
145] one can find non-quotient hyperrings as well.
Therefore, we know 4 different classes of hyperfields, so far: the class of the quotient hyperfields and the three ones which are constructed via the Theorems 11, 12 and 13.
The open and closed hypercompositions [
5] in the hyperfields are of special interest. Regarding these, we have the following:
Proposition 27. In a hyperfield the sum of any two non-opposite elements x, y ≠ 0 does not contain the participating elements if and only if, the difference equals to , for every .
Proposition 28. In a hyperfield the sum of any two non-opposite elements contains these two elements if and only if, the difference equals to , for every .
For the proofs of the above Propositions 27 and 28, see [
141]. With regard to Proposition 28, it is worth mentioning that there exist hyperfields in which, the sum
contains only the two addends
, i.e.
, when
and
[
141].
Theorem 14. [
141]
Let be a hyperfield. Let be a hypercomposition on , defined as follows:Then, is a hyperfield and moreover, if is a quotient hyperfield, then is a quotient hyperfield as well. Corollary 10. If is a field, then is a quotient hyperfield.
The following problem in field theory is raised from the study of the isomorphism of the quotient hyperfields to the hyperfields which are constructed with the process given in Theorem 14:
when does a subgroup G of the multiplicative group of a field F have the ability to generate F via the subtraction of G from itself? [
141,
143]
A partial answer to this problem, which is available so far, regarding the finite fields is given with the following theorem:
Theorem 15. [
146]
Let F be a finite field and G be a subgroup of its multiplicative group of index and order . Then, G-G = F, if and only if:,
and ,
, and ,
, and ,
, and ,
, and ,
, and .
Closely related to the hyperfield is the hypermodule and the vector space.
Definition 9. A left hypermodule over a unitary hyperring is a canonical hypergroup with an external composition , from to satisfying the conditions:
- i.
- ii.
- iii.
- iv.
and
for all and all .
The right hypermodule is defined in a similar way. A hypermodule over a hyperfield is called vector hyperspace.
Suppose
and
are hypermodules over the hyperring
. The cartesian product
can become a hypercompositional structure over
, when the operation and the hyperoperation, for
,
, and
, are defined componentwise, as follows:
The resulting hypercompositional structure is called the direct sum of and .
Theorem 16. The direct sum of the hypermodules is not a hypermodule.
Proof. Let
and
be two hypermodules over a hyperring
. Then:
On the other hand:
Therefore:
Consequently axiom (ii) is not valid. □
Remark 2. Errors in Published Papers. Unfortunately, there exist plenty of papers which incorrectly consider that the direct sum of hypermodules is a hypermodule. For instance, they mistakenly consider that if is a hyperring or a hyperfield, then is a hypermodule or a vector hyperspace over respectively. Due to this error, a lot of, if not all the conclusions of certain papers are incorrect. We are not going to specifically refer to such papers, as we do not wish to add negative citations in our paper, but we refer positively to the paper by Ρ. Ameri, M Eyvazi and S. Hoskova-Mayerova [120], where the authors have presented a counterexample which shows that the polynomials over a hyperring give a superring in the sense of Mittas [118,119] and not a hyperring, as it is mistakenly mentioned in a previously published paper which is referred there. This error can also be highlighted with the same method as the one in Theorem 16, since the polynomials over a hyperring can be considered as the ordered sets where , i=0, 1, … are their coefficients. Following the above remark, we can naturally introduce the definition:
Definition 10. A left weak hypermodule over a unitary hyperring is a canonical hypergroup with an external composition , from to satisfying the conditions (i), (iii), (iv) of the Definition 9 and, in place of (ii), the condition:
ii’. for all and all .
The quotient hypermodule over a quotient hyperring is constructed in [
147], as follows:
Let
be a
−module, where
is a unitary ring, and let
be a subgroup of the multiplicative semigroup of
, which satisfies the condition
, for all
. Note that this condition is equivalent to the normality of
only when
is a group, which appears only in the case of division rings (see [
144]). Next, we introduce in
the following equivalence relation:
After that, we equip
with the following hypercomposition, where
is the set of equivalence classes of
modulo ~:
i.e.,
consists of all the classes
which are contained in the set-wise sum of
. Then
becomes a canonical hypergroup. Let
be the quotient hyperring of P over G. We consider the external composition from
to
defined as follows:
This composition satisfies the axioms of the hypermodule and so
becomes a
- hypermodule.
If
is a module over a division ring
, then, using the multiplicative group
of
we can construct the quotient hyperring
and the relevant quotient hypermodule
. For any
it holds that
. In [
147] it is shown that this hypermodule is strongly related to the projective geometries. A. Connes and C. Consani, in [
139,
140] also prove that the projective geometries, in which the lines have at least four points, are exactly vector hyperspaces over the quotient hyperfield with two elements. Moreover, if
is a vector space over an ordered field
, then, using the positive cone
of
we can construct the vector hyperspace
over the quotient hyperfield
. In [
147] it is shown that every Euclidean spherical geometry can be considered as a quotient vector hyperspace over the quotient hyperfield with three elements.
Modern algebraic geometry is based on abstract algebra which offers its techniques for the study of geometrical problems. In this sense, the hyperfields, were connected to the conic sections via a number of papers [
148,
149,
150], where the definition of an elliptic curve over a field
F was naturally extended to the definition of an elliptic hypercurve over a quotient Krasner hyperfield. The conclusions obtained in [
148,
149,
150] were extended to cryptography as well.